I've tried connecting direct from the airport time machine to the ethernet socket on my I Mac. I keep getting, no cable connection? I've also tried direct from the modem to the airport T M, Same result. I'm running out of options and need help

If you are talking about a Macbook Pro, then yes because it has 2 thunderbolt ports on the left side. If you mean just a Macbook, then no as you will need an adapter to connect from USB-C to gigabit ethernet.

This adapter is not necessary to connect your iMac to your Time Capsule for internet access and for your Time Machine backups. Time Capsule and iMac can be connected via Wifi. In almost all cases, though, using this adapter with a decent ethernet cable (I recommend at least a Cat 5e ethernet cable) connected hard-wired to your Time Capsule will not only improve your internet speed on your iMac, but also your Time Machine backup speed. The actual improvement in speed of these things will depend on your current internet speed and distance between the iMac and Time Capsule along with other possible interferences. In most cases, though, I would say the average user would notice a slight increase in performance, and the power user to notice a significant performance increase.

If you deal with backup up many large files and are a heavy-bandwidth user, I woul definitely recommend using this adapter with either a Cat 5e or Cat 6 cable.